Here are Ben Franklin's two famous magic squares which (together with
their lesser-known counterparts) are discussed in my paper "The Lost
Squares of Dr. Franklin," American Mathematical Monthly, June-July 2001.
An entirely different approach is described in my new book on Franklin,
due out soon! A hint can be found in "A Bent For Magic,"
Mathematics Magazine, Feb. 2006.

Instead of requiring diagonal sums to be constant (as in a fully
magic square), Franklin used "bent rows" such as those highlighted
below.

The construction I will give owes much to the various papers which
attempted
to decode Franklin's method [Carus 1906, Marder 1940, Chandler 1951, Patel
1991]. However, these approaches were not concerned with his actual
motivation
for performing each step. The recipe given below includes speculation on
why these particular steps were chosen.

We are not concerned here with the proof that the resulting square is
Franklin magic, since that has been outlined elsewhere [Günther 1876,
Abrahams
1994] and does not appear to give any idea of Franklin's original
reasoning. It is important to keep in mind that the author of the squares
was unschooled in number theory.

For simplicity of notation, the 8-square will be our pictorial model,
but the procedure outlined here (and its concomitant motivation) applies
to any 8n by 8n matrix.

The order of the square we seek is 8n, so the entries are 1,
2, 3, …, 64n2. The square was built two columns at a
time. Certainly if those column pairs have constant row sum and constant
column sum, then the resulting square will be semi-magic.
(Each column pair is an 8n by 2 "magic rectangle," in the
parlance
of our times.)

To this end, we will partition the set {1, 2, 3, …,
64n2}
into 4n subsets of equal size. Since magic squares often pair
complementary
entries, as when they are associated (associative), it makes
sense for these subsets to do so. In this square they will be paired in
a novel way, but the idea of pairing associated entries in some
fashion was an old one, even in the 1700's.

Proceeding naively/intuitively, the simplest such partition is: {1,
2, ... , n} and their complements {64n2,
64n2
-
1,
... , 64n2 -
n
+ 1};
next {n + 1, n + 2, ..., 2n} and their complements;
etc. (Here and in later steps it is instructive to compare the "lost"
Franklin
16-square from Section 9 of my paper. The strategy there was quite
different,
but it did proceed by forming row pairs in a like fashion.)

If we simply write the elements of the first subset in order, filling
2 columns, we get:

8

57

7

58

6

59

5

60

4

61

3

62

2

63

1

64

The row sum is constant, but not the column sum. That is easily
remedied:

57

8

7

58

59

6

5

60

61

4

3

62

63

2

1

64

We can proceed likewise for the next column pair, filling it with
entries
from the next subset: {n + 1, n + 2, ..., 2n} and
{64n2- n,
64n2
-
n-
1,
... , 64n2 -
2n
+ 1};
the resulting rectangle will also be magic. But before doing so, consider:
the rectangle pictured above can be transformed in many ways without
destroying
its magic properties. Thus, if at the end our procedure results in a
square
which is is only semi-, not Franklin, magic, there are many other options
left to try. One such transformation, the one Franklin used, is shown
below:

57

8

7

58

59

6

5

60

61

4

3

62

63

2

1

64

57

8

7

58

59

6

5

60

61

4

3

62

63

2

1

64

61

4

3

62

60

5

6

59

58

7

8

57

63

2

1

64

This is not to say that Franklin proceeded by trial-and error, only
that if he did so it was after conveniently narrowing the possibilities.

For the next partition-set, we proceed similarly:

16

49

15

50

14

51

13

52

12

53

11

54

10

55

9

56

49

16

15

50

51

14

13

52

53

12

11

54

55

10

9

56

49

16

15

50

51

14

13

52

53

12

11

54

55

10

9

56

52

13

14

51

53

12

11

54

55

10

9

56

50

15

16

49

Why build that 2nd column pair in a slightly different fashion? This
step will be justified in a moment. First, we will combine the work so
far. One option would be to place the two column pairs side by side, thus:
and that would certainly be "magic" in the rows and columns. Indeed, this
step appears in the construction of the "lost" 16-square, using rows
instead
of columns. However, we have another option: "nest" the column pairs:
Observe
now that the step from n to n + 1 (8 to 9 in our picture)
is reminiscent of another magic square technique, namely the "Lo Shu" and
its extensions to higher odd orders. There, one follows a knight's-move
scheme until there's no room left, then proceeds to an adjacent cell and
continues. This simplest of magic square techniques must have been known
to Franklin. Perhaps the slight change in strategy between column pair
1 and column pair 2 is due to a reasoning be analogy: I have run out of
room, so I will move over to the adjacent square.

Nesting those column pairs results in this magic rectangle:

52

14

53

11

55

9

50

16

61

4

3

62

60

5

6

59

58

7

8

57

63

2

1

64

13

51

12

54

10

56

15

49

Continuing in this manner, one obtains a Franklin magic square with
all of the desired properties:

36

45

52

61

4

13

20

29

30

19

14

3

62

51

46

35

37

44

53

60

5

12

21

28

27

22

11

6

59

54

43

38

39

42

55

58

7

10

23

26

25

24

9

8

57

56

41

40

34

47

50

63

2

15

18

31

32

17

16

1

64

49

48

33

However, it is not yet the example Franklin gave, which is:

Nevertheless, it is a simple matter to bridge the gap. For, these last
two
squares
have the following interesting property: if one moves an even
number of
columns from the left-hand side to the right, not only are the row and
column sums preserved (obviously), it is also true that the "bent row"
sums are preserved as well. For some of the bent rows this is obvious,
but for others this is a marvelous additional quality Franklin did not
describe. Surely he was aware of it, though, because that is the final
step in his construction; moving the first 2 columns in our square from
left side to right completes the procedure for drawing the 8-square. In
the 16-square and in the 8n-square this is also his
final step: move 2n columns from left to right.

The aforementioned property is also a crucial element in the magic
circle construction, given in my paper "The Lost Squares of
Dr.
Franklin." It implies that V-shaped and ^-shaped bent rows can be
translated, not just up and down, but also right or left (by an even
number of cells) without disturbing the magic sum. (In the magic
circle, this property is responsible for 10 of the 20 "excentric
circles".) Although Franklin never remarked
on this property explicitly in his letters, its significant role in the
constructions of both the
squares and the circle indicates that he knew of it.